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Create QR Code JIS X 0510 in Java in the form S = m dX dX .

in the form S = m dX dX .
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SOLUTION Let s start by recalling that X 2 = X X . So we can rewrite the action S in the following way: S = = 1 1 2 2 d a X m a 2 m2 1 X2 d X 2 m2 2 2 X2 m m2 1 d X2 m X2 2 X2 m2 1 2 d X 2 X m X X 2
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String Theory Demysti ed
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Now, let s use a simple algebraic trick to rewrite the rst term. Remember from complex variables that i 2 = 1. This means that m2 2 m2 2 X X = ( 1)( 1) X2 X2 = = m2 2 2 i X X2 m2 4 4 i X X2
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= m 2i 4 X 2 = m i 4 X 2
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But i 4 = +1 , and so m i 4 X 2 = m X 2 = m X X . Therefore the action is m2 1 d X 2 m X X 2 X2 1 d m X X m X X 2
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S = =
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= m d
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X X = S
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This demonstrates that the two actions are equivalent.
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Strings in Space-Time
At this point we have reviewed some basic techniques that help us calculate the equations of motion for a free relativistic point particle. We are going to extend this work to the case of a string moving in space-time. A point particle has no extent whatsoever, so can be described as a zero-dimensional object. We have seen that its
CHAPTER 2 Equations of Motion
The world-sheet of a closed string is a tube in space-time.
motion can be described by saying that a point particle (zero-dimensional) sweeps out a path or line in space-time (one dimension) that we call the world-line. A string, unlike a point particle, has some extension in one dimension, so it s a onedimensional object. As it moves, the string (one-dimensional) sweeps out a twodimensional surface in space-time that scientists call a worldsheet. For example, imagine a closed loop of string moving through space-time. The worldsheet in this case will be a tube, as shown in Fig. 2.1. We can summarize this in the following way: The path of a point particle is a line in space-time. A line can be parameterized by a single parameter, which is the proper time. As a string moves through space-time it sweeps out a two-dimensional surface called a worldsheet. Since the worldsheet is two-dimensional, we need two parameters, which we can generally denote by 1 and 2 . Locally the coordinates 1 and 2 can be thought of as coordinates on the worldsheet. Or, another way to look at this is to parameterize the worldsheet, we need to account for proper time and the spatial extension of the string. So, the rst parameter
String Theory Demysti ed
is once again the proper time , and the second parameter, which is related to the length along the string, is denoted by :
1 =
2 =
Coordinates on the worldsheet ( , ) are mapped onto space-time by the functions (called the string coordinates) X ( , ) (2.12)
So time and spatial position on the string are mapped onto the spatial coordinates in (d + 1) dimensional space-time as {X 0 ( , ), X 1 ( , ), , X d ( , )} Now, we need to write down the action for the string which will generalize Eq. (2.8) to our new higher-dimensional world, that is, to the case of the worldsheet. This is done in the following way. Recall that the action of a point particle is proportional to the length of its world-line [Eq. (2.5)]. We just noted that a string sweeps out a two-dimensional worldsheet in space-time. This tells us that if we are going to generalize the notion of the action of a point particle, we might expect that the action of a string is proportional to the surface area of the worldsheet. This is in fact the case. Anticipating that the constant of proportionality will turn out to be the string tension, we can write this action as S = T dA (2.13)
where dA is a differential element of area on the worldsheet. To nd the form of dA, we start by considering a differential line element ds 2 and introduce coordinates on the worldsheet as 1 = and 2 = . Doing a little algebra we have ds 2 = dX dX = X X d d
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